$(3^{25} + 3^{26} + 3^{27} + 3^{28})$ is divisible by

### Concept & Formula The core strategy here is to factor out the term with the smallest exponent. By applying the distributive property of exponents, we can simplify the expression into a product of a prime power and an integer, which will clearly reveal its divisors. $$x^a + x^{a+1} + x^{a+2} = x^a(1 + x + x^2)$$ ### Step-by-Step Solution * **Given:** The expression $(3^{25} + 3^{26} + 3^{27} + 3^{28})$. * **Factor out the smallest power:** Extract $3^{25}$ as a common factor from all terms. $$3^{25} (1 + 3^1 + 3^2 + 3^3)$$ * **Evaluate the bracketed sum:** $$3^{25} (1 + 3 + 9 + 27)$$ $$3^{25} (40)$$ * **Decompose to match options:** The options are 11, 16, 25, 30. We need to see if $3^{25} \times 40$ contains one of these. Let us borrow a $3$ from $3^{25}$: $$3^{24} \times 3 \times 40$$ $$3^{24} \times 120$$ * **Deduction:** Since 120 is exactly divisible by 30 ($120 \div 30 = 4$), the entire expression is divisible by 30. ### Exam Strategy & Shortcut Whenever you see a sum of consecutive powers, immediately factor out the lowest power. Calculate the numerical value of the remaining bracket. The answer will almost always be a factor of that bracketed sum, or a factor of the bracketed sum multiplied by the base (in this case, $40 \times 3 = 120$, which reveals 30). ### Common Pitfall Students often incorrectly try to add the bases or exponents directly (e.g., thinking $3^{25} + 3^{26} = 6^{51}$), which violates fundamental exponent rules. Always factor instead of adding. ### Final Answer Therefore, the correct answer is **30**.